Why a Minimum Variance Portfolio Is Unique
Summary
The document asks for a canonical reference establishing uniqueness of the minimum variance portfolio, defined as minimizing portfolio variance under linear constraints. It gives the key mathematical condition: when the covariance matrix is positive definite, the variance objective is strictly convex, and the feasible set defined by linear constraints is convex. A strictly convex objective over a convex feasible set has at most one minimizer, so the solution is unique when a feasible minimizer exists.
The discussion points to a standard convex optimization textbook for the proof and notes that older optimization literature may contain earlier academic references. A response also distinguishes unconstrained short selling, where an analytical solution may be available, from constrained cases that may require numerical optimization. It cautions that practical minimum variance portfolios depend heavily on estimating variances and covariances accurately; uniqueness of the mathematical solution does not guarantee reliable portfolio weights.
Key ideas
- A positive definite covariance matrix makes portfolio variance a strictly convex objective.
- Linear portfolio constraints define a convex feasible set.
- Strict convexity on a convex feasible set implies at most one minimum variance solution.
- Portfolio uniqueness does not resolve the practical difficulty of estimating the covariance matrix.
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# What is the canonical reference for Minimum Variance Portfolio's uniqueness?
# What is the canonical reference for Minimum Variance Portfolio's uniqueness?
I am writing a white paper in which I am trying to compare a strategy to different well-known - and classic - asset allocation optimization approaches.
One of the methods I chose is the minimum variance portfolio $w_\text{MV}$ defined as follows:
$$w_\text{MV} = \underset{w}{\arg \min} ~ w' \Sigma w$$
where $\Sigma$ is the covariance matrix of the assets and under the linear constraints $Aw \leq b$ and $E w = d$.
I have always heard that the MV portfolio was unique, and I know that this problem is linked to quadratic programming which I believe guarantees a unique solution as long as $\Sigma$ is positive-definite.
I wanted to add a reference to another paper where this uniqueness was discussed (proved), and I found several ones written quite recently. However, I was wondering if there was one paper thas was more famously known for discussing that particular property?
## Answer by Marc Shivers (score 3)
https://quant.stackexchange.com/a/5978
For academic references, you will likely have to look in the very early optimization literature.
Uniqueness of the MV portfolio follows immediately from the lemma that a strictly convex function on a convex set has no local minima.
The standard textbook reference is Convex Optimization by Boyd and Vandenberghe. See section 4.2.2 in particular. A free online copy is available at stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf.
## Answer by vonjd (score 2)
https://quant.stackexchange.com/a/4572
This article by Eric Falkenstein is exactly what you are looking for:
Early Low Vol Literature Now Everywhere
EDIT Falkenstein has a new post out on the academic origins of the approach: Here
## Answer by jianpan (score 1)
https://quant.stackexchange.com/a/4569
If short sell is allowed, I remember there's a unique analytical solution, otherwise it has to be solved numerically. Is your approache different? IMHO the issue of min variance approach is really not how to solve this constrained optimization problem, but how to estimate asset return and var/covar matrix accurately.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.